# Theorems

• Jan 4th 2007, 03:03 PM
pyrogurl6989
Theorems
1)Use the remainder theorem and the factor theorem to determine whether (b+4) is a factor of (b^3 +3b^2 - b+12)

A. The remainder is 0 and, therefore, (b+4) is a factor of (b^3 +3b^2 - b+12)
B. The remainder is 0 and, therefore, (b+4) isn't a factor of (b^3 +3b^2 - b+12)
C. The remainder isn't 0 and, therefore, (b+4) is a factor of (b^3 +3b^2 - b+12)
D. The remainder isn't 0 and, therefore, (b+4) isn't a factor of (b^3 +3b^2 - b+12)

2) Use the remainder theorem to determine the remainder the remainder when 3t^2 +5t -7 is divided by t-5

3)Use the remainder theorem and the factor theorem to determine whether (c+5) is a factor of (c^4 +7c^3 +6c^2 -18c+10)

A. The remainder is 0 and, therefore, (c+5) is a factor of (c^4 +7c^3 +6c^2 -18c+10)
B. The remainder is 0 and, therefore, (c+5) isn't a factor of (c^4 +7c^3 +6c^2 -18c+10)
C. The remainder isn't 0 and, therefore, (c+5) is a factor of (c^4 +7c^3 +6c^2 -18c+10)
D. The remainder isn't 0 and, therefore, (c+5) isn't a factor of (c^4 +7c^3 +6c^2 -18c+10)
• Jan 4th 2007, 08:03 PM
Soroban
Hello, pyrogurl6989!

I will assume you know the Remainder Theorem and the Factor Theorem.

Quote:

1) Use the remainder theorem and the factor theorem to determine
whether $\displaystyle b+4$ is a factor of $\displaystyle f(b) \:=\:b^3 +3b^2 - b+12$

Since $\displaystyle f(\text{-}4)\;=\;(\text{-}4)^3 + 3(\text{-}4)^2 - (\text{-}4) + 12\;=\;-64 + 48 + 4 + 12 \;=\;0$

. . then: .(A) The remainder is 0, and therefore $\displaystyle b+4$ is a factor of $\displaystyle f(b).$

Quote:

2) Use the remainder theorem to determine the remainder
when $\displaystyle f(t)\:=\:3t^2 +5t -7$ is divided by $\displaystyle t-5$

Since $\displaystyle f(5)\:=\:3(5^2) + 5(5) - 7 \:=\:93$, then the remainder is $\displaystyle 93.$

Quote:

3)Use the remainder theorem and the factor theorem to determine
whether $\displaystyle c+5$ is a factor of $\displaystyle f(c) \:=\:c^4 +7c^3 +6c^2 -18c+10$

Since $\displaystyle f(\text{-}5) \;=\;(\text{-}5)^4 + 7(\text{-}5)^3 + 6(\text{-}5)^2 - 18(\text{-}5) + 10 \;=\;625 - 875 + 150 + 90 + 10 \;=\;0$

. . then: .(A) The remainder is 0, and therefore $\displaystyle c+5$ is a factor of $\displaystyle f(c)$